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So for practice
with expectation,

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let's calculate the expected
number of heads in n coin

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flips, and just working
directly from the definition,

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because we have
tools to do that.

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So we're imagining n independent
flips of a coin with bias p.

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So the coins might not be fair.

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The probability of heads is p.

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It would be biased in favor of
heads if p is greater than 1/2

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and biased against heads
if p is less than 1/2.

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And we want to know how
many heads are expected.

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This is a basic question that
will come up again and again

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when we look at random variables
and probability theory.

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So what's the expected
number of heads?

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Well, we already know--
we've examined the binomial

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distribution B n,p.

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B n,p is telling us how
many heads there are in n

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independent flips.

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So we're asking about the
expectation of the binomial

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variable B n,p.

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Well, let's look
at the definition.

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The definition of B n,p is it's
the sum over all the possible

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values of B, namely all
the numbers from 0 to n--

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that's k-- of the probability
of getting k heads.

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And this formula here
is the probability

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of getting k heads, which
we've worked out previously.

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n choose k times p to the k,
1 minus p to the n minus k.

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Well, let's introduce
an abbreviation,

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a standard abbreviation.

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Let's replace 1 minus p by q,
where-- so p plus q equals 1,

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and they're both not
negative and between 0 and 1.

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And when I express the
expectation this way,

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it starts to look like
something a little bit familiar.

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And our strategy is going to
be to use the binomial theorem,

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and then the trick
of differentiating it

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is going to wind up
giving us a closed

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formula for this expression
for the expectation

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of the binomial random variable.

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So let's remember the
binomial theorem says

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that the nth power of x
plus y is the sum all from k

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equals 0 to n of n choose k, x
to the k, y to the n minus k.

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And if I differentiate
this, what

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happens is that on
the left hand side,

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if I differentiate with
respect to x, I get x plus y

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to the n minus 1 times n.

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And if I differentiate the
right hand side-- let's

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differentiate it term by term.

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And differentiating
with respect to x

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is going to turn this n choose
k x to the k, y to the n minus k

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into an x to the k
minus 1 times k term.

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But I'd like to keep
the n-- the k here

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and the k there matching.

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So that after
differentiating, that

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becomes an x to the k minus 1.

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Let's multiply it by x
to make it x to the k.

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And of course, I have to undo
that multiplication by dividing

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the whole thing by 1/x.

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So by differentiating
the binomial formula,

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we get the following formula
for this sum that is starting

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to look just like the
expectation of B n,p,

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1/x times the sum from k equals
0 to 1 of k times n choose k,

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x to the k, y to the n minus k.

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Well, let's compare
the two terms.

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So here's this term
and there's this one.

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I'm going to replace this line
by the formula for expectation

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of the binomial random variable.

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So this is what we're
trying to evaluate,

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and I have this great theorem.

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You can see how they match up.

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So what I'm going to do is
replace p and q-- replace

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x and y in this
general formula that I

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got by differentiating the
binomial theorem with p and q.

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And what happens?

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So I just plug in the p and q.

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Now, the left hand
side. p plus q is 1.

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So the left hand side
is going to become n.

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And this right hand side now
is exactly the expectation of B

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n,p-- this part of it, anyway.

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So what I'm going to wind up
with is that n is equal to 1/p

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times the expectation of B n,p.

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In other words, the expectation
of B n,p is n times p,

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and that is the basic formula
that we were deriving by first

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principles without using
any general properties

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of expectation, just the
definition of expectation

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and the stuff that we had
already worked out in terms

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of the binomial theorem.